Let \(d\)  be an odd positive square-free integer and \(\zeta _8\)  a primitive 8-th root of unity. Note that the triquadratic field \(L_d=\mathbb {Q}(\zeta _8,\sqrt{d})\)  is the first step of
both the cyclotomic \(\mathbb {Z}_2\) -extension and a non-cyclotomic \(\mathbb {Z}_2\) -extension (e.g. the non-cyclotomic \(\mathbb {Z}_2\) -extension of
\(\mathbb {Q}(\sqrt{-1})\)  (cf. [1]}) shifted by \(\sqrt{d}\) ) of the biquadratic field
\(\mathbb {Q}(\sqrt{-1},\sqrt{d})\) . This fact makes it of a particular importance from the triquadratic number fields.
In this subsection and the next section, we will make some preparations on the 2-class groups and unit groups of some multiquadratic number fields that will help to study the 2-class field towers and the second 2-class groups of all fields \(L_d=\mathbb {Q}(\zeta _8,\sqrt{d})\) , such that
2-class group is of type \((2, 2)\) , together with some different families of mutiquadratic number fields in the last section of this paper.
For this, we recall the following results of our earlier paper [2]}.
Let \(p\) , \(p^{\prime }\)  and \(q\)  be three prime integers. The 2-class group of \(L_d=\mathbb {Q}(\sqrt{d}, \zeta _8)\)  is of type \((2, 2)\)  if and only if \(d\)  takes one of the following forms:
\(d=pq \text{ with } p\equiv -q\equiv 1\pmod {4}, \left(\dfrac{2}{p}\right)=-1, \left(\dfrac{2}{q}\right)=1 \text{ and } \left(\dfrac{p}{q}\right)=-1.\) 
\(d=pq \text{ with } p\equiv q\equiv -1\pmod {4}, \left(\dfrac{2}{p}\right)=-1, \left(\dfrac{2}{q}\right)=1 \text{ and } \left(\dfrac{p}{q}\right)=-1 ,\) 
\(d=p^{\prime }\text{ with } p^{\prime }\equiv 1 \pmod {16} \text{ and } \left(\frac{2}{p^{\prime }}\right)_4\ne \left(\frac{p^{\prime }}{2}\right)_4 .\) 
